Space-time fractional diffusions in Gaussian noisy environment

Abstract

This paper studies the linear stochastic partial differential equation of fractional orders both in time and space variables , where is a general Gaussian noise and , . The existence and uniqueness of the solution, the moment bounds of the solution are obtained by using the fundamental solutions of the corresponding deterministic counterpart represented by the Fox H-functions. Along the way, we obtain some new properties of the fundamental solutions.

Keywords:

• Gaussian noisy environment
• time fractional diffusions
• time and space fractional order SPDE
• Fox H-functions
• fundamental solutions
• nonnegativity
• heat kernel type estimate
• chaos expansion
• moment estimates

Acknowledgements

Le Chen thanks Francesco Mainardi for some useful discussion and thanks Anatoly N. Kochubei for pointing out Pskhu’s paper [31].

Notes

No potential conflict of interest was reported by the authors.

Dedicated to Prof. Bernt Øksendal on the occasion of his 70th birthday.

Additional information

Funding

L. Chen is partially supported by the Swiss National Foundation for Scientific Research fellowship [P2ELP2_151796]; Y. Hu is partially supported by a grant from the Simons Foundation [#209206]; J. Huang is supported by the National Science Foundation [grant number DMS-1440140] while in residence at the Mathematical Research Institute in Berkeley, California, during the fall 2015 semester.